Giant g factor tuning of long-lived electron spins in Ge

Control of electron spin coherence via external fields is fundamental in spintronics. Its implementation demands a host material that accommodates the highly desirable but contrasting requirements of spin robustness to relaxation mechanisms and sizeable coupling between spin and orbital motion of charge carriers. Here we focus on Ge, which, by matching those criteria, is rapidly emerging as a prominent candidate for shuttling spin quantum bits in the mature framework of Si electronics. So far, however, the intrinsic spin-dependent phenomena of free electrons in conventional Ge/Si heterojunctions have proved to be elusive because of epitaxy constraints and an unfavourable band alignment. We overcome such fundamental limitations by investigating a two dimensional electron gas (2DEG) confined in quantum wells of pure Ge grown on SiGe-buffered Si substrates. These epitaxial systems demonstrate exceptionally long spin relaxation and coherence times, eventually unveiling the potential of Ge in bridging the gap between spintronic concepts and semiconductor device physics. In particular, by tuning spin-orbit interaction via quantum confinement we demonstrate that the electron Land\'e g factor and its anisotropy can be engineered in our scalable and CMOS-compatible architectures over a range previously inaccessible for Si spintronics.

Spin-orbit interaction (SOI) couples charge and spin degrees of freedom 1 . This effect has sparked considerable interest because it results in a suitable spin splitting even in the absence of external magnetic fields. SOI governs spin-dependent phenomena such as Bychkov-Rashba physics 2,3,4 , persistent spin helix states 5,6,7 , spin Hall 8,9,10 and spin Seebeck effects 11,12 , offering novel and exciting perspectives for utilizing spin currents in non-magnetic materials 13 . This opens viable avenues for the end-of-the-roadmap implementation of semiconductor spintronics 14 .
The Landé factor describes the susceptibility of the spin state of a charge carrier to an external field and sets a key metrics for the strength of SOI in the solid state framework. In Si, the fundamental building block of mainstream microelectronics, the weak SOI manifests itself as a negligible deviation of the electron factor from the isotropic free carrier value 0 ≈ 2 15 . Spinorbit coupling due to bulk inversion asymmetry is in fact absent in Si due to its centrosymmetric crystal structure 1,16 . Seminal works demonstrating tailoring of SOI in Si rather focused on lowdimensional Si/SiGe heterosystems, in which SOI becomes more important at the interfaces as a result of the induced spatial inversion asymmetry 17,18,19 . Yet the factor tunability in such systems remained very small 20 .
In this contest, we turned our attention to Ge because it exhibits a highly anisotropic factor 21 due to a stronger SOI than the lighter Si, while it shares with Si the key prerequisites for any practical implementation of quantum information processing, namely a long spin relaxation time and a substantial abundance of spin-less isotopes 22,23 . In view of its full compatibility with the technology of integrated circuits and its exceptionally high bulk mobility, Ge is also increasingly seen as a viable option for replacing Si in conventional low-power logics 24 and can thus be regarded as an attractive candidate for transport in novel spintronic architectures. made from superconductors and self-assembled nanocrystals 27 , while core-shell Ge/Si nanowires 28 have been envisioned as hosts of Majorana fermions 29 .
To date, however, efforts have been mainly focused on the spin physics of holes. Besides the large lattice mismatch, which induces growth defects and poor material and interface quality, the spontaneous type II band alignment at Ge/Si heterojunctions 30,31 has so far precluded the experimental study of SOI of conduction electrons confined in Ge. Indeed charge carriers are spatially separated by the built-in potential, which favours holes (electrons) at the Ge (Si) side of the heterointerface.
In light of the pivotal advances reported in the field of Si photonics 32,33 , we expect that band-gap engineering in SiGe alloys will similarly provide tremendous advantages to semiconductor spintronics by opening unexplored pathways for the full exploitation of Ge. The vast degrees of freedom offered by strain and alloying in dictating the band-edge offsets in SiGe heterostructures motivated us to design n-type modulation (n-mod) doped devices on Si that consist of pure Ge quantum wells (QW) embedded in Ge-rich SiGe barriers with a 10 nm thick phosphorous doped region at their centre (Fig.1a). The individual layers were engineered in order to obtain negligible strain with respect to the SiGe buffer, as confirmed by high resolution x-ray diffraction (HRXRD) measurements summarized in Fig.1b,c and in the Supplementary Section A. Such straincompensation accommodates the compressed QW within tensely strained barriers and precludes the formation of additional defects at the interfaces. The resulting Ge/SiGe heterojunction allows us to gather direct access to a type I band alignment, with a notable accumulation of L-valley electrons (see Fig.1d and Fig.2a) in the Ge well due to a robust confining potential of the order of 60 meV. This, combined with conduction electron spin resonance (CESR), permits experimental manipulation of the electron factor theoretically predicted in Ge more than a decade ago 34 . To this end, we carried out a systematic study of samples that, according to HRXRD, differ by the QW thickness, namely 201, 171 and 161 nm.
We found that the low temperature cyclotron resonance (CR) strongly depends on the relative orientation of an external magnetic field B with respect to the sample surface. As shown in Fig.2b, the sample with the largest width of the QWs and without remote doping does not show a CR signal when B lies along the [110] direction (in-plane field). On the other hand, when B is rotated towards the [001] growth direction (perpendicular field), the spectrum exhibits a very pronounced signal.
The resonance field decreases as θ → 0 according to 1/cosθ (see Supplementary Section B) 17 . This behaviour is a clear signature that carriers are confined in the (001) plane, where they can undergo cyclotron motion driven by the electric field of the microwave. This well-defined CR and its characteristic dependence upon illumination, discussed in detail in the Supplementary Section B, provides direct proof of the existence of a 2DEG in the QWs plane 17 , and the absence of low temperature localization of carriers on impurity sites.
As a consequence, we have direct access to the intrinsic spin-dependent properties of conduction electrons. This constitutes a remarkable difference with respect to previous electron spin resonance studies applied to Ge. Aside works focussed on electrons bound to donors, very few experiments suggested the peculiar presence of an ESR due to delocalized electrons in antimony-doped bulk Ge at low temperatures 35,36,37 . Such finding was ascribed to partial population of conduction band states by the built-in inhomogeneous strain fields randomly experienced by electrons at different Sb sites. Instead, our heteroepitaxial n-mod architecture naturally guarantees itinerant electrons in the Ge layer and their concomitant spatial separation from the remote donors, that reside in the SiGe barrier.
This point is further corroborated by the following results. In addition to the CR signal, Fig.2c shows that four well-resolved CESR peaks become prominent in n-mod samples. These peaks (A to D in Fig.2c) strongly shift in their spectral position when increasing the angle  between B and the normal to the sample surface from 0° to 90°. This dependence demonstrates a highly anisotropic factor, as summarized in Fig.2d. Notably, we did not succeed in observing peaks corresponding to electrons localized on P donors, neither in the SiGe barriers nor in the Ge wells. The origin of the narrow resonance lines shown in Fig.2c and of their marked angular dispersion can be rationalized as detailed below.
In bulk Ge each conduction band edge at the four equivalent L points of the Brillouin zone has an ellipsoidal energy surface oriented along a 111 crystal direction (see Fig.2a). According to Roth and Lax 21 , the factor matrix of free electrons reflects such spheroidal shape and its C 3v symmetry 35 . Hence for any angle  between the external field and the major axis of one ellipsoid of revolution, the concomitant effective value of can be obtained as follows: where and are the two independent parallel and transverse components lying along or being normal to the major axis of the ellipsoid, respectively. Here is the angle between B and the major axis of the ellipsoid 38 . preserve the bulk C 3v symmetry of the -tensor. Such finding is in sharp contrast to the behaviour of the magneto-conductivity tensor, which rules the CR response ( Fig.2b). This might be a consequence of the fact that the latter is mostly determined by heterointerface properties, while the factor deviations from the free electron value are caused first of all by SOI 17 . A theoretical approach will be mentioned below. It is worth noting that the observation of a well-resolved CESR multiplet proves that spin relaxation of conduction electrons in QWs is dominated by zone-centre intravalley rather than zone-edge intervalley electron-phonon coupling 40 . The latter due to scattering among the different L minima would have otherwise averaged out the factors, eventually yielding a single CESR line 41 . We emphasize that the inversion symmetry of the Ge lattice is well-known to preclude D'yakonov-Perel type spin-flip processes so that spin relaxation is essentially mediated by Elliott-Yafet mechanisms. This feature and the unique SOI experienced by thermal electrons at the conduction band edge of Ge have been recently suggested resulting in exceptionally long-lived electron spin states 40,42 . By working at cryogenic temperatures, we could selectively quench the intervalley scattering, that previous literature work recognized as a major factor in limiting the experimentally accessible spin relaxation times 39,43 . Such approach opens up a largely unexplored scenario offering the possibility to capture ultimate spin-flip and dephasing mechanisms.   In particular, we notice that to a first approximation the experimental data can be recovered by applying a rigid shift of 2.9 G to the calculated ∆ (compare solid and dotted lines in Fig.4b). Since this offset points towards isotropic decoherence mechanisms, we suggest the following scenario to explain the physics leading to such 2.9 G broadening.  Fig.4b can thus be accounted for by the two aforementioned opposing effects, namely hyperfine coupling and motional narrowing. In our Ge QWs the prominent role of the latter leads to a remnant hyperfine broadening of 2.9 G. This is substantially narrower than the 10 G resonance linewidth of electrons fully bound to shallow donors that is well known for bulk samples with natural isotopic abundance of 73 Ge 49,22 .
It shall be noted that the phenomenon discussed above neglects broadening due to spin-flip processes, consistently with the spin relaxation times addressed in the following.
Data in the upper panel of Fig.4b further show the occurrence of slightly different linewidths at the same value, thus suggesting the presence of additional, albeit weaker, dephasing mechanisms.
With this respect, it is illuminating to note that we measured two peaks: One at θ ~ 35° (Fig.2d), After having discussed all the mechanisms contributing to the observed CESR linewidth, we can determine the relaxation time of the spin ensemble 2 * , which provides a lower limit for the spin decoherence time 2 48 , as follows: where ħ is the reduced Planck constant, the Bohr magneton, is obtained from the CESR peak, and Δ 0 can be obtained by the following relation 52 : shall be noted that in the latter case the spin decoherence time was found to be anisotropic, reflecting the intervalley scattering regime 39 . Fig.4b demonstrates that when the intravalley relaxation is the dominating process the ensemble dephasing time is not -factor dependent and thus isotropic.
In the following we will try to extract the spin-lattice or longitudinal relaxation time 1 from the power ( ) dependence of continuous wave ESR 48 . To this end, we carried out selected measurements in a cylindrical cavity with high Q factor and offering a finite electric field of the microwave within the sample. Moreover, we restricted ourselves to the analysis of CESR lines at = , because, as shown before, those are unaffected by the inhomogeneous broadening induced by the interface roughness. Fig.5a shows a colour-coded map of the CESR intensity as a function of in the −30 dB (low ) to −7 dB (high ) range for the resonance peak corresponding to degenerate C and D valleys measured at  = 90° in the sample having 17 nm thick QWs. Besides readily demonstrating that ∆ remains constant, Fig.5a shows that at low the CESR signal possesses the well-known absorption lineshape (AS), which results from spin-flip processes induced by the resonance between the microwave photons and the Zeeman splitting of the spin states. For a direct inspection, the CESR peak measured at −30dB is shown as a black line in the inset in Fig.5b. Surprisingly, however, we do not recover in our samples the typical increase and saturation with of the AS intensity that is routinely observed in electron spin resonance experiments 48 . Instead, a more puzzling behaviour can be appreciated in Fig.5a. At low the lineshape resembles the well-studied showing one unexpected negative dip, which stems from a pure DS (see also inset of Fig.5b).
Notably, by further increasing the intensity of the resonance peak turns out to be strongly